# on an interval

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##### 1: 1.4 Calculus of One Variable
If $f(x_{1})\leq f(x_{2})$ for every pair $x_{1}$, $x_{2}$ in an interval $I$ such that $x_{1}, then $f(x)$ is nondecreasing on $I$. … If also $f(x)$ is continuous on the right at $x=a$, and continuous on the left at $x=b$, then $f(x)$ is continuous on the interval $[a,b]$, and we write $f(x)\in C[a,b]$. … If $f(x)$ is continuous on an interval $I$ save for a finite number of simple discontinuities, then $f(x)$ is piecewise (or sectionally) continuous on $I$. … If $f^{(n)}$ exists and is continuous on an interval $I$, then we write $f\in C^{n}(I)$. …
##### 2: 26.2 Basic Definitions
Unless otherwise specified, it consists of horizontal segments corresponding to the vector $(1,0)$ and vertical segments corresponding to the vector $(0,1)$. For an example see Figure 26.9.2. … A partition of a set $S$ is an unordered collection of pairwise disjoint nonempty sets whose union is $S$. … A partition of a nonnegative integer $n$ is an unordered collection of positive integers whose sum is $n$. As an example, $\{1,1,1,2,4,4\}$ is a partition of 13. …
##### 3: 28.17 Stability as $x\to\pm\infty$
However, if $\Im\nu\neq 0$, then $(a,q)$ always comprises an unstable pair. …
##### 4: 18.36 Miscellaneous Polynomials
These are OP’s on the interval $(-1,1)$ with respect to an orthogonality measure obtained by adding constant multiples of “Dirac delta weights” at $-1$ and $1$ to the weight function for the Jacobi polynomials. …
##### 5: Bibliography K
• R. B. Kearfott (1996) Algorithm 763: INTERVAL_ARITHMETIC: A Fortran 90 module for an interval data type. ACM Trans. Math. Software 22 (4), pp. 385–392.
• ##### 6: 18.2 General Orthogonal Polynomials
It is assumed throughout this chapter that for each polynomial $p_{n}(x)$ that is orthogonal on an open interval $(a,b)$ the variable $x$ is confined to the closure of $(a,b)$ unless indicated otherwise. (However, under appropriate conditions almost all equations given in the chapter can be continued analytically to various complex values of the variables.) … More generally than (18.2.1)–(18.2.3), $w(x)\,\mathrm{d}x$ may be replaced in (18.2.1) by a positive measure $\,\mathrm{d}\alpha(x)$, where $\alpha(x)$ is a bounded nondecreasing function on the closure of $(a,b)$ with an infinite number of points of increase, and such that $0<\int_{a}^{b}x^{2n}\,\mathrm{d}\alpha(x)<\infty$ for all $n$. … All $n$ zeros of an OP $p_{n}(x)$ are simple, and they are located in the interval of orthogonality $(a,b)$. …
##### 7: 10.23 Sums
provided that $f(t)$ is of bounded variation (§1.4(v)) on an interval $[a,b]$ with $0. …
##### 8: 1.6 Vectors and Vector-Valued Functions
$\mathbf{c}(t)=(x(t),y(t),z(t))$, with $t$ ranging over an interval and $x(t),y(t),z(t)$ differentiable, defines a path. … with $(u,v)\in D$, an open set in the plane. …
##### 9: 2.3 Integrals of a Real Variable
$k$ ($\leq\infty$) and $\lambda$ are positive constants, $\alpha$ is a variable parameter in an interval $\alpha_{1}\leq\alpha\leq\alpha_{2}$ with $\alpha_{1}\leq 0$ and $0<\alpha_{2}\leq k$, and $x$ is a large positive parameter. …
##### 10: 18.1 Notation
 $x,y$ real variables. … weight function $(\geq 0)$ on an open interval $(a,b)$. …